Emergence of collective oscillations in adaptive cells

Shou-Wen Wang1, 2, 3, ∗ and Lei-Han Tang1, 4, 5, †

1Beijing Computational Science Research Center, Beijing, 100094, China2Department of Engineering Physics, Tsinghua University, Beijing, 100086, China3Department of Systems Biology, Harvard Medical School, Boston, MA 02115, USA

4Department of Physics and Institute of Computational and Theoretical Studies,Hong Kong Baptist University, Hong Kong, China

5State Key Laboratory of Environmental and Biological Analysis,Hong Kong Baptist University, Hong Kong, China

(Dated: November 23, 2019)

AbstractCollective oscillations of cells in a population appear under diverse biological contexts. Here, weestablish a set of common principles by categorising the response of individual cells against a time-varying signal. A positive intracellular signal relay of sufficient gain from participating cells isrequired to sustain the oscillations, together with phase matching. The two conditions yield quan-titative predictions for the onset cell density and frequency in terms of measured single-cell andsignal response functions. Through mathematical constructions, we show that cells that adapt to aconstant stimulus fulfil the phase requirement by developing a leading phase in an active frequencywindow that enables cell-to-signal energy flow. Analysis of dynamical quorum sensing in severalcellular systems with increasing biological complexity reaffirms the pivotal role of adaptation inpowering oscillations in an otherwise dissipative cell-to-cell communication channel. The physicalconditions identified also apply to synthetic oscillatory systems.

IntroductionHomogeneous cell populations are able to exhibit a richvariety of organised behaviour, among them periodic os-cillations. During mound formation of starved socialamoebae, cyclic AMP waves guide migrating cells to-wards the high density region1–5. Elongation of the ver-tebrate body axis proceeds with a segmentation clock6,7.Multicellular pulsation has also been observed in nervetissues8, during dorsal closure in late stage drosophilaembryogenesis9–12, and more13. In these examples, com-munication through chemical or mechanical signals is es-sential to activate quiescent cells. Dubbed dynamicalquorum sensing (DQS) to emphasise the role of increasedcell density in triggering the auto-induced oscillations,this class of behaviour lies outside the well-known Ku-ramoto paradigm of oscillator synchronisation14,15.

Interestingly, auto-induced oscillations have also beenreported in situations without an apparent biologicalfunction. A case in point is otoacoustic emission (OAE),where a healthy human ear emits sound spontaneously ina silent environment16,17. Anatomically, sound is gener-ated by hair bundles, the sensory units of hair cells thatdetect sound with ultra-high sensitivity18–20. Anotherexample is glycolytic oscillations of yeast cells which canbe induced across different laboratory conditions21–26.This type of phenotypic behaviour may not confer bene-fits to the organism, so their existence is puzzling.

Here, we consider a population of cells attempting tomodulate temporal variations of the extracellular concen-tration of a protein or analyte, or a physical property of

∗ Correspondence: shouwen wang@hms.harvard.edu† Correspondence: lhtang@csrc.ac.cn

their environment, by responding to it. The response ofa cell to the external property, or signal, can be medi-ated by an arbitrary intracellular biochemical network.By focusing on the frequency-resolved cellular response,we report a generic condition for collective oscillations toemerge, and show that it is satisfied when cells affect thesignal in a way that adapts to slow environmental vari-ations, i.e., cells respond to signal variation rather thanto its absolute level. In particular, we prove the exis-tence of an active frequency regime, where adaptive cellsanticipate signal variation and attempt to amplify thesignal. Sustained collective oscillations emerge when acell population, beyond a critical density, communicatesspontaneously through such a channel.

We provide a physical explanation of oscillations interms of energy driven processes, with adaptive cells out-putting energy in the active frequency regime upon stim-ulation. For mechanical signals, the energy output is di-rectly observable as work on the environment. For chem-ical signals, chemical free energy is transferred during therelease of molecules into the extracellular medium. To-gether with the measurable response of individual cells,quantitative predictions of the oscillation frequency andits dependence on cell density become possible.

The adaptive cellular response highlighted in thiswork is shown to underlie several known examplesof DQS, and possibly glycolytic oscillations in yeastcell suspensions. The ubiquity of adaptation27–37 inbiology may also explain the emergence of inadvertentoscillations. We discuss implications and predictionsof this general mechanism at the end of the paper, inconnection with previous experimental and modellingwork.

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FIG. 1. Spontaneous oscillations in a communicating cell pop-ulation. (A) The scenario of mechanical oscillations wherecells communicate via a shared displacement s of the physicalenvironment. Activity a of a cell against the displacement isregulated by a hidden intracellular network which respondsto s through a mechanical sensor. (B) Illustration of chemi-cal oscillations where cells interact via a shared extracellularsignal s. The signal is sensed and secreted by individual cells.

ResultsNecessary conditions for auto-induced oscilla-tions. We begin by considering a scenario of mechan-ical oscillations, as illustrated in Fig. 1A. Later we willshow that the same results hold for chemical oscillations.The cells are spatially close enough so that they could beregarded as under the same environment. Here, the ex-tracellular signal s is taken to be the deformation of themechanical environment, which is both sensed and mod-ified by participating cells. The cell activity that affectsthe environment is denoted by a variable a. Its dynam-ics is controlled by an unspecified intracellular regulatorynetwork that responds to s through a mechanical sensor.To see how the intracellular activities might disrupt sta-sis in an equilibrium state of s, we consider the followingLangevin equation:

γs = F (s) +N∑j=1

α1aj + ξ. (1)

Here γ is the friction coefficient, F (s) the external forcethat tries to restore the physical environment, ξ the ther-mal noise, and the sum represents the total force createdby N active cells in a unit volume, whose strength is setby α1 > 0. In general, the cell activity depends on thepast history of the signal. Upon a small change of s, theaverage response of the activity of jth cell satisfies

〈aj(t)〉 = 〈aj〉u +

∫ t

−∞Raj

(t− τ)〈s(τ)〉dτ, (2)

with 〈·〉 and 〈·〉u denoting noise average with and with-out an external time-varying signal, respectively. With-out loss of generality, we set the stationary activity 〈aj〉uto zero. The activity response function Ra is a prop-erty of the intracellular molecular network, which can becomputed for specific models38,39 or measured directlyin single-cell experiments3,4,19,40. In general, Ra may de-pend on the ambient signal level s of the cell.

The shared signal s offers a means to synchronisethe activities of cells. We derive here a matching con-dition for s and the a’s to enter a positive signal re-lay. Expressing Equation (2) in Fourier form, we have

〈aj(ω)〉 = Raj(ω)〈s(ω)〉. For weak disturbances, the

restoring force in Equation (1) can be approximated bya linear one, i.e., F (s) ' −Ks. Consequently, 〈s(ω)〉 =∑N

j=1 α1Rs(ω)〈aj(ω)〉, where

Rs =1

K − iγω(3)

is the signal response function, with i the imaginary unit.For identical cells, these equations yield an oscillatory so-lution a(ωo) 6= 0 provided Nα1Ra(ωo)Rs(ωo) = 1. Togain more insight, we express the two response func-tions in their amplitudes and phase shifts, i.e. Ra ≡|Ra| exp(−iφa) and Rs ≡ |Rs| exp(−iφs). Then, the celldensity N = No and the selected frequency ωo at theonset of collective oscillations are determined by,

φa(ωo) = −φs(ωo), (4)

|Ra(ωo)Rs(ωo)| = (α1No)−1. (5)

These are essentially conditions of linear instability forthe quiescent state expressed in terms of the single-cell and signal response functions, and constitute ourfirst main result. For inhomogeneous cell populations,one simply replaces Ra by its population average Ra ≡N−1

∑Nj=1Raj

.Under the assumption of additive signal release from

individual cells as expressed by Equation (1), we nowhave a mathematical prediction for the onset density No

and oscillation frequency ωo. Let α2 ∼ |Ra| be the sensi-tivity of the cell activity against s. We introduce a signalrelay efficiency N ≡ Nα1α2, which also sets the couplingstrength of cellular activities through the signal. Oscilla-tions start at the critical coupling strength No = Noα1α2.Equation (5) simply states that, at the selected frequencyωo, signal amplification through the collective action ofNo cells compensates signal loss from dissipative forcesacting on s, e.g., friction for a mechanical signal or degra-dation/dilution for a chemical signal. The frequency ωo

is chosen such that phase shifts incurred in the forwardand reverse medium-cell transmissions match each other[Equation (4)].

Although we mainly focus on the emergence of collec-tive oscillations as cell density increases, oscillation deathat high cell densities through a continuous transition,

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FIG. 2. Dynamical response of an intracellular adaptive vari-able a. (A) Response to a stepwise signal: after a transientresponse, a returns to its pre-stimulus state (within a smallerror ε). In the simplest case, the transient response is con-trolled by the activity shift timescale τa and the circuit feed-back timescale τy. Solid and dashed lines correspond to over-damped (τa � τy) and under-damped (τa � τy) situations,respectively. (B) Response to a sinusoidal signal at low (left)and high (right) frequencies. The phase shift φa switches sign.

as observed in certain experimental systems, may alsofulfil the self-consistency condition [Equation (4-5)]. Anexample is provided in Supplementary Note 6 (see alsoSupplementary Figure 15), where, due to the nonlinearproperties of the intracellular circuit, the cell activitybecomes less responsive at elevated signal intensity. Tomake use of our procedure, the cell-density dependenceof the linear response functions needs to be considered.

Cell-to-signal energy flow. Auto-induced collectiveoscillations must be driven by intracellular active pro-cesses. These active components of the system give anonequilibrium character to the activity response41–44

and furthermore enable energy flow from the cell to thesignal upon periodic stimulation, an interesting physicalphenomenon left unnoticed so far.

To set the stage, we turn to basic considerations ofnon-equilibrium thermodynamics45,46. The shared sig-nal s as illustrated in Fig. 1 typically follows a dissipa-tive dynamics such as Equation (1). When the medium isclose to thermal equilibrium, the Fluctuation-DissipationTheorem (FDT) relates the imaginary component R′′s of

the signal response Rs to its spontaneous fluctuation Cs

induced by thermal noise38,39,47: 2TR′′s (ω) = ωCs(ω),

where Cs(ω) = 〈|s(ω)|2〉u is the spectral amplitude ofthe signal, and T is the temperature. This relation de-mands R′′s (ω) to be positive at all frequencies. Hence,the dissipative nature of the physical environment trans-lates into a phase delay, i.e., φs ≡ − arg(Rs) ∈ (−π, 0).Under the over-damped signal dynamics [Equation (1)],Equation (3) gives

φs(ω) ≡ − arg(Rs(ω)

)= − tan−1(ωτs) ∈

(−π

2, 0), (6)

where τs = γ/K is the signal relaxation time. (The sit-uation −π < φs(ω) < −π/2 occurs at high frequencieswhen the dynamics of s is underdamped.) On the otherhand, a leading phase as required by Equation (4) for theintracellular signal relay, violates the FDT. In the presentcase, active cells play the role of the out-of-equilibrium

partner. We have calculated the work done by one ofthe cells on the signal when the latter oscillates at a fre-quency ω (see Supplementary Note 1). The output power

W ≡ 〈s ·α1a〉, i.e., the averaged value of the product be-tween signal velocity (s) and force from an individual cell(α1a), is given by

W ' −α1ωR′′a(ω)〈|s(ω)|2〉

= α1ω|Ra(ω)| sinφa(ω)〈|s(ω)|2〉. (7)

The energy flux is positive, i.e., flowing from the cell tothe signal, when a has a phase lead over s, re-affirmingEquation (4) as a necessary condition on thermodynamicgrounds. Stimulated energy release from an active cell tothe signal as expressed by Equation (7) constitutes oursecond main result in this paper.

Equation (7) can also be used to calculate the energyflux for an arbitrary signal time series s(t), providedthe linear response formula Equation (2) applies. Inparticular, thermal fluctuations of s in the quiescentstate may activate a net cell-to-signal energy flow. Thetotal power is obtained by integrating contributions fromall frequencies. Previous experiments from Hudspethlab yielded a phase-leading response of hair bundles tomechanical stimulation at low frequencies19. The samegroup also showed that energy can be extracted fromthe hair bundle via a slowly oscillating stimulus18.

Chemical oscillations. The criteria given byEquation (4-5) apply equally to chemical oscillationsillustrated in Fig. 1B. In contrast to the mechanicalsystem, Equation (1) at γ = 1 becomes a rate equationfor the extracellular concentration s of the signallingmolecules. The term F (s) (negative) gives the degrada-tion or dilution rate of s in the medium, while individualcells secrete the molecules at a rate proportional totheir activity a. As the signalling molecules are con-stantly produced and degraded, chemical equilibrium isoften violated even in the steady state. Nevertheless,F (s) usually plays the role of a stabilising force so

that the signal response function Rs(ω) has the samephase-lag behaviour as the mechanical case. Releaseof the molecules by the communicating cells must bephase-leading so as to drive oscillatory signalling.

Adaptive cells show phase-leading response.Apart from the aforementioned hair bundles, phase-leading response to a low frequency signal has also beenreported in the activity of E. coli chemoreceptors40 andin the osmo-response in yeast36. Interestingly, all threeof these cases are examples of adaptive sensory systemswhose response to a step signal at t = 0 is shown inFig. 2A. The small activity shift ε at long times is knownas the adaptation error. Fig. 2B shows the response ofthe same system under a sinusoidal signal. The low fre-quency response exhibits a phase lead while the high fre-quency one has a phase lag. Below, we show that thesign switch in the phase shift of an adaptive variable isan inevitable consequence of causality.

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FIG. 3. A weakly nonlinear model with adaptation. (A-C)Single cell response. (A) A noisy two-component model withnegative feedback. (B) Frequency-resolved phase shift φa =

− arg(Ra). A sign change takes place at ω = ω∗ ' (τaτy)−1/2,

with a leading s on the low frequency side. (C) Real (R′a) and

imaginary (R′′a) components of the response spectrum. R′a is

of order ε in the zero frequency limit, while R′′a changes signat ω = ω∗. Also shown is the correlation spectrum Ca(ω)multiplied by ω/(2T ), where T is the noise strength. The

fluctuation-dissipation theorem R′′a = ωCa(ω)/(2T ) for ther-mal equilibrium systems is satisfied on the high frequencyside, but violated at low frequencies. (D-F) Simulations ofcoupled adaptive circuits. (D) Time traces of the signal(red) and of the activity (blue) and memory (cyan) from oneof the participating cells at various values of the couplingstrength N = α1α2N . (E) The oscillation amplitude A (ofactivity a) and frequency ω against N . The amplitude A

grows as (N − No)1/2 here, a signature of Hopf bifurcation.(F) Determination of oscillation frequency from the renor-malised phase matching condition at finite oscillation ampli-tudes: φ+

a (ω,A) = −φ+s (ω,A). The linear model for s yields

φ+s (ω,A) = −φs(ω). Parameters: τa = τy = γ = K = c3 = 1,α1 = α2 = 0.5, and ε = 0.1. The strength of noise terms isset at T = 0.01.

From the causality condition Ra(t < 0) = 0, the real

(R′a) and imaginary (R′′a) part of the response function infrequency space satisfy the Kramers-Kronig relation48:

R′a(ω) =2

π

∫ ∞0

R′′a(ω1)ω1

ω21 − ω2

dω1. (8)

For a step signal of unit strength, Equation (2) yields

ε = 〈a(∞)〉 − 〈a〉u =

∫ ∞0

Ra(τ)dτ = limω→0

R′a(ω). (9)

Comparing Eqs. (8) and (9) in the limit ω → 0 and as-

suming ε to be sufficiently small, we see that R′′a(ω) in-side the integral must change sign. In other words, bothphase-leading (R′′a < 0) and lagged (R′′a > 0) behaviourare present across the frequency domain. This is ourthird result.

Adaptation plays a key role in biochemical net-works27,28, and especially in sensory systems20,29,31–36.Connection between adaptation and collective oscilla-tions has been implicated in previous works4,49,50. Withthe mathematical results presented above, the pathwayfrom adaptation to phase-leading response, and ontocollective oscillations through signal relay, is firmlyestablished (See Methods). Below we illustrate, withthe help of three examples of increasing complexity, howthis line of reasoning could link up different aspects ofsystem behaviour to arrive at a renewed understanding.Implications of our model study to experimental workare given in the Discussion section.

A weakly nonlinear model with adaptation. Weconsider first a noisy two-component circuit which is avariant of the model for sensory adaptation in E. coli41

(Fig. 3A, see also Methods). For weak noise, the intra-cellular signal relay in the quiescent state is essentiallylinear with the receptor response function given by

Ra(ω) = α2

[1+

ε

ε2 + (τyω)2+iτaω

∗ (ω∗/ω)− (ω/ω∗)

1 +(ε/(τyω)

)2 ]−1,(10)

where

ω∗ = (τaτy)−1/2(1− ε2τa/τy)1/2. (11)

Here τa and τy are the timescales for the activity (a)and negative feedback (y) dynamics, respectively. InFigs. 3B,C, we show the phase shift φa(ω) and the real

and imaginary part of Ra(ω) against the frequency ω,plotted on semi-log scale. As predicted, φa(ω) under-goes a sign change at ω∗. Correspondingly, the imagi-nary component of the response R′′a becomes negative inthe phase-leading regime, violating the FDT. The peakof |Ra(ω)| is located close to ω∗, with a relative width∆ω/ω∗ ' Q−1 where Q = τaω

∗ ' (τa/τy)1/2.Allowing the chemoreceptor activity a to affect the sig-

nal as in Equation (1) with F (s) = −Ks, we observean oscillatory phase upon increase in cell density in nu-merical simulations (Fig. 3D). Fig. 3E shows the oscilla-tion amplitude (upper panel) and frequency (lower panel)against the coupling strength N around the onset of oscil-lations. The threshold coupling strength No = Noα1α2

and the onset frequency ωo both agree well with the val-ues predicted by Equation (4-5) (see arrows in Fig. 3E).The transition is well described by a supercritical Hopf

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bifurcation. At finite oscillation amplitudes, there is adownward shift of the oscillation frequency which can bequantitatively calculated in the present case by introduc-ing a renormalised response function R+

a (ω) (see Supple-mentary Note 2), whose phase is shown in Fig. 3F. Theoscillation frequency is determined by the crossing of thetwo curves φs(ω) and φ+a (ω), with the formal indepen-dent of the oscillation amplitude A. As the oscillationamplitude grows further, higher order harmonics gen-erated by the nonlinear term become more prominent.The model system eventually exits from the limit cyclethrough an infinite-period bifurcation and arrives at anew quiescent state. The upper bifurcation point Nb isinversely proportional to the adaptation error ε (see Sup-plementary Figure 1).

The signal phase shift φs(ω) is given by Equation (6).When the signal relaxation time τs is much shorter thanthe cell adaptation time τ∗ ≡ 2π/ω∗, φs(ω) stays close tozero so that the selected period is essentially given by τ∗.In this case, |Ra(ω)| is near its peak and hence the celldensity required by Equation (5) is the lowest. As signalclearance slows down, the crossing point shifts to lowerfrequencies. Given a finite adaptation error ε > 0, thereis a generic maximum signal relaxation time τ∗s ∼ ε−1

beyond which the phase matching cannot be achieved(see Supplementary Note 3 and also SupplementaryFigure 3).

Excitable dynamics. DQS in Dictyostelium and othereukaryotic cells takes the form of pulsed release ofsignalling molecules2,7,51. The highly nonlinear two-component FitzHugh-Nagumo (FHN) model is often em-ployed for such excitable phenomena3,52,53. Similar tothe sensory adaptation model discussed above, each FHNcircuit has a memory node y that keeps its activity alow (the resting state) under a slow-varying signal s(t)(Fig. 4A, see also Methods). On the other hand, a suf-ficiently strong noise fluctuation or a sudden shift ofs sends the circuit through a large excursion in phasespace (known as a firing event) when y is slow (i.e.,τy � τa). Our numerical investigations show that thenoise-triggered firing does not disrupt the adaptive na-ture of the circuit under the negative feedback from y.The noise-averaged response of a single FHN circuit ex-hibits the same characteristics as the sensory adaptationmodel, including adaptation to a stepwise stimulus aftera transient response (Fig. 4B, upper panel), as well as thephase-leading behaviour and diminishing response ampli-tude on the low frequency side (Fig. 4B, lower panel).

Fig. 4C shows time traces of individual cell activities(blue and green curves) as well as that of the signal s(red curve) from simulations of weakly coupled FHNcircuits at three different values of the coupling strengthN (see Methods). At N = 0.5, the two selected cells fireasynchronously while s remains constant. At N = 0.9,collective behaviour as seen in the oscillation of s startsto emerge, although individual circuits continue to firesporadically. Upon further increase of N , synchronised

FIG. 4. Simulations of the coupled excitable FitzHugn-Nagumo (FNH) model with noise. (A) Model illustration.Note the self-activation of a that gives rise to excitability (seeMethods for details). (B) Noise-averaged response of a in theresting state. Upper panel: the average response to a stepsignal. Lower panel: the response amplitude and phase shiftat various signal frequencies. (C) Trajectories of the coupledFHN model at various values of the effective coupling strengthN . In addition to the signal s, activities of two out of a to-tal of 1000 cells are plotted. (D) Signal oscillation amplitudeand frequency against effective cell density. Red stars: simu-lation data; Blue circles: predictions of Equation (4-5) usingnumerically computed response spectra.

firing is established. Despite the highly nonlinear natureof the FHN model, both the onset coupling strength No

and the frequency ωo are well predicted by Equation (4-5) using the respective response functions in the restingstate (Fig. 4D).

Yeast glycolytic oscillations. We take the adapt-to-oscillate scenario one step further to examine the dy-namics of ATP autocatalysis in yeast. Concentrationoscillations of NADH and glycolytic intermediates havebeen observed in yeast cell extracts as well as in starvedyeast cell suspensions upon shutting down the respira-tory pathway (see Ref.22 for a review). The phospho-fructokinase (PFK), an enzyme in the upper part of theglycolytic pathway, is tightly regulated by ATP, a keyproduct of glycolysis. This robust negative feedback iscommonly regarded as the driver of glycolytic oscilla-tions, with a typical period of 30-40 seconds in intact cellsbut 2 minutes or longer in extracts. Cells at high densityoscillate synchronously due to redox signalling via thefreely diffusing molecule acetaldehyde (ACE)22,26. As thecell density decreases, the synchronised behaviour breaksdown. While many studies found continued oscillation of

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FIG. 5. Yeast glycolytic oscillations. (A) The reaction network of glycolysis in a yeast cell (see Supplementary Figure 3 forfull names of the abbreviations). (B) Single-cell phase diagram spanned by the intracellular glucose and acetaldehyde (ACE)concentrations. Adaptation becomes less accurate further away from the oscillatory region (white), as indicated by colouredbands. The response of ATP changes sign around ACEin,0 ' 0.2 as indicated by the dashed line. (C) Representative time tracesof metabolites to an upshift in intracellular ACE concentration (ACEin) for selected conditions in each band (triangle, star, anddiamond in B). (D) Frequency-resolved phase shifts of selected metabolites to weak sinusoidal perturbations of ACEin. ATP,BPG and PEP are phase-leading in both blue and orange bands of the phase diagram, while PYR does so only in the blueband. (E) A reduced model for glycolytic oscillations where the intracellular NAD/NADH ratio and pyruvate (PYR) act as thereceiver and sender of the signal (ACE), respectively. Adaptive response of PYR to ACE is coupled to the homeostasis of ATPthrough the reaction PYK. (F) Phase diagram of the reduced model against intracellular ACE concentration. (G) Oscillationamplitudes (upper panel) and time-averaged intracellular and extracellular ACE concentrations (lower panel) against celldensity in a population where individual cells metabolise according to the reduced model. Data for three selected values ofACE membrane permeability D are shown.

individual cells at their own frequencies25,54, simultane-ous disappearance of individual and collective oscillationsas in other DQS systems has also been reported23. Weshow below that both type of behaviour could be accom-modated in a model of glycolysis that couple intracellularredox state to ATP autocatalysis.

We first investigate the dynamic properties of single-cell glycolysis at fixed intracellular glucose and ACE con-centrations. Collective oscillations in yeast cell suspen-sions, which require ACE transport across the cell mem-brane, will be discussed later. Our starting point isthe du Preez et al. model55 that includes around 20metabolic reactions (Fig. 5A). By monitoring the tem-poral response of metabolites under perturbations of theintracellular ACE concentration, we obtained a phase di-agram shown in Fig. 5B. The white region marks spon-taneous oscillations in an isolated cell56. The glucoseconcentration, which controls glycolytic flux, needs to besufficiently high for oscillations to take place. ACE alsohas a role in the dynamics: either very low or very highconcentrations arrest the oscillations.

The non-oscillatory part of the phase diagram can befurther divided into 3 sub-regimes according to the adap-

tive properties of the metabolic network against an ACEsignal (coloured bands in Fig. 5B). Fig. 5C gives, underone representative condition in each band, the concentra-tion variation of four metabolites upon a sudden shift inthe intracellular ACE concentration ACEin. In all cases,the intracellular redox agent NAD follows closely ACEconcentration change and hence acts as an instantaneoustransducer of the signal. ATP adapts in both the or-ange and blue band, while PYR, the substrate to produceACE, adapts only in the blue band. TRIO, the metabo-lite immediately upstream of the reaction GAPDH thatuses NAD as cofactor, does not adapt. Overall, the adap-tation error increases progressively as one moves awayfrom the oscillatory region. A sign change in the re-sponse of ATP (and also of TRIO) takes place across thedashed line at ACEin,0 ' 0.2 mM.

Fig. 5D shows phase shifts of ATP, NAD and five othermetabolites along the glycolytic pathway against a peri-odic ACE signal at various frequencies. At the pointmarked by star in the orange band, ATP, BPG and PEPare phase-leading (after a π shift) below the frequencyω∗ ' 20 min−1 (upper panel). The list is expandedto all six metabolites (except NAD which is synchro-

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nised with the signal) when the environment shifts tothe point marked by diamond in the blue band (lowerpanel). In particular, PYR (blue line) have a large phaselead around ω∗, which matches its adaptive behaviourunder Fig. 5C (Supplementary Note 4).

To further disentangle dynamical properties of the net-work, we constructed a reduced model in Fig. 5E by tak-ing into account stoichiometry and known regulatory in-teractions along the glycolytic pathway24, and by makinguse of the timescale separation in the turnover of metabo-lites as suggested by their response spectra (Fig. 5D andSupplementary Note 5). Since ATP and PYR now ap-pear as co-products of the condensed reaction PYK inthe reduced model, the latter can be viewed as a re-porter of ATP homeostasis implemented by the negativefeedback loop (cyan line in Fig. 5E). Fig. 5F shows thephase diagram of the reduced model against the intracel-lular ACEin,0. Similar to the full model at high glucoseconcentrations, the circuit enters an oscillatory state atintermediate values of ACEin,0, and shows adaptive re-sponse on the two wings. The extended adaptive regimeon the high ACEin,0 side differs from the behaviour seenin Fig. 5B, but is reproduced by a mutant of the fullmodel where the glyoxylate shunt (GLYO) is turned off(Supplementary Figures 10-11).

We now examine a model of yeast cell suspensionswhere individual cells metabolise according to thereduced model and communicate their redox statethrough ACE (Supplementary Note 6). ACE is syn-thesised internally and degraded at rates kin and kexwithin and outside the cell, respectively. The rate ofits cross-membrane transport is set by the membranepermeability D. When D is large, the intracellularand extracellular ACE levels are essentially the same(left panel, Fig. 5G). On the low density side, thehomogeneous cell population enters the oscillatoryphase through synchronisation of oscillatory cells. Thisbehaviour continues beyond the point ACEin = 0.72(arbitrary unit) when an isolated cell switches fromoscillation to adaptation (Fig. 5F). In other words, thesystem crosses over smoothly from oscillator synchroni-sation to adaptation-driven oscillations, or DQS. As Ddecreases and becomes comparable to ACE degradationrates (set at kin = 0.5 and kex = 0.3), the intracellularACE concentration grows and eventually exceeds theupper self-oscillatory threshold ACEin = 0.72 even atthe low cell density limit. Our simulations indicate thatDQS persists at D = 0.4 but disappears at D = 0.1(middle and right panels, Fig. 5G). Interestingly, DQSat D = 0.4 disappears at an upper threshold densityρc2 = 0.73. This inverse DQS, i.e., oscillation quenchingat high cell density, can be quantitatively explainedby Equations (4-5) using the numerically determinedresponse functions that depend on the cell density(Supplementary Note 6 and Supplementary Figure 15).

DiscussionIn this work, we investigated a general scenario for emerg-

ing oscillations in a group of cells that communicate viaa shared signal. It covers a broad class of pulsation be-haviour in cell populations, collectively known as dynam-ical quorum sensing. Using the single-cell response to ex-ternal stimulation, we formulated a quantitative require-ment for the onset of collective oscillations that must besatisfied by active cells as well as models of them. A proofis presented to link this requirement to the adaptive re-lease of signalling molecules by individual cells. Our workthus consolidates observations made in the literature andformalises adaptation as a unifying theme behind DQS.

The above mathematical results connect well to the re-cent surge of interest in active systems, where collectivephenomena emerge due to energy-driven processes on themicroscopic scale57,58. The study of such non-equilibriumprocesses opens a new avenue to explore mechanisms ofspontaneous motion on large scales. We presented a gen-eral formula for the energy outflow of a living cell througha designated mechanical or chemical channel under pe-riodic stimulation. This energy flux is positive over arange of frequencies when the cell responds to the stim-ulus adaptively. Since adaptation is a measurable prop-erty of a cell, the thermodynamic relation is applicablewithout making specific assumptions about intracellularbiochemical and regulatory processes, while most modelsdo. When cells are placed together in a fixed volume, aquorum is required to activate the energy flow via selfand mutual stimulation.

We reported three case studies to illustrate how thesegeneral yet quantitative relations could be applied toanalyse the onset of collective oscillations in specific cellpopulations. Our first example is a coarse-grained modelwhere signal reception and release are integrated into thesame activity node (e.g., a membrane protein or a molec-ular motor). Due to the weak nonlinearity of the intracel-lular circuit, many analytical results were obtained. Theintracellular adaptive circuit has two timescales: the ac-tivity relaxation time τa and the negative feedback timeτy. Their ratio Q2 = τa/τy, similar to the quality fac-tor in resonators, determines the shape of the adaptiveresponse (Fig. 2). At small adaptation error ε � 1, the

imaginary part of the response function Ra(ω) changessign at the characteristic frequency ω∗ ' (τaτy)−1/2.

This is also approximately the frequency where |Ra(ω)|reaches its maximum. When cells are coupled throughthe signal with a relaxation time τs, the onset oscillationfrequency ωo increases with decreasing τs, reaching itsmaximal value ω∗ when τs � 1/ω∗.

Much of these results carry over to our second exam-ple, a population of coupled excitable circuits describedby the FitzHugh-Nagumo model. Despite its highly non-linear nature, the FHN model in the resting state showsadaptive response under weak stimulation. Our numeri-cal simulations of the coupled system at weak noise con-firm the onset oscillation frequency and the critical celldensity predicted by Equation (4-5).

The above theoretical predictions compare favourablywith available experimental data. The first is mechani-

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FIG. 6. Intracellular activity response function constructedfrom single-cell measurements on Dictyostelium. (A) Upperpanel: The average cytosolic cAMP level (the activity a, ar-bitrary unit) in response to a step increase of 1 nM extra-cellular cAMP at t = 0 (reproduced from Fig. 2A in Ref.3).Lower panel: The response function Ra(t) estimated from thederivative of the response data in the upper panel. (B) Thecorresponding phase shift φa in the low frequency regime (redcurve), obtained from the Fourier transform of Ra(t). Onsetoscillation frequencies in experiments reside in the green re-gion (see Fig. 2B in Ref.2). Also shown is the phase shift−φs from Equation (6) at τs = 0.35 min (blue curve), theextracellular signal clearance time to yield an onset oscilla-tion period of about 8 min. Faster signal clearance shortensthe oscillation period towards the theoretical lower bound ataround 6.28 min.

cal stimulation of hair cells carried out by Martin et al.19,where the cellular response was extracted using a flexi-ble glass fibre. Deformation of the glass fibre, which isthe signal here, has a relaxation timescale (∼ 0.5 ms)much shorter than the adaptation time of the hair bun-dle (∼ 0.1 s). Spontaneous oscillations of the combinedsystem were observed at 8 Hz, the predicted frequencywhere the imaginary part of the hair bundle responsefunction R′′a(ω) undergoes the expected sign change. Thesecond is a recent microfluidic single-cell measurementof Dictyostelium reported by Sgro et al.3, where thechange of cytosolic cAMP level (activity a) in responseto extracellular cAMP variation (signal s) was presented.From the measured response a(t) to a step increase ofthe signal in their work (reproduced in Fig. 6A, upperpanel), we computationally deduced the response func-tion Ra(t) = da/dt in the time domain (Fig. 6A, lower

panel) and then the response spectrum Ra via Fouriertransform. The resulting phase shift φa changes signaround ω∗ = 1 min−1 (Fig. 6B). According to our theory,the onset oscillation period at high flow rate should bearound 6.28 min, which is indeed what was observed inexperiments2–4.

DQS in Dictyostelium is a time-dependent phe-nomenon coupled to cell migration and development1,5.In the experiments reported in Refs.2,3, synchronised fir-ing of cells starts five hours after nutrient deprivation.The period of firing shortens from 15 - 30 min at theonset to 8 min and thereafter 6 min as cells begin toaggregate. Therefore the onset of collective oscillationsmay not be triggered by a critical cell density as such

but the cell density does affect the period of oscillations.Previously, a property known as fold-change detection(FCD) was invoked and verified to explain cell-cell sig-nalling even when cells are far apart4. In FCD, the in-tracellular signal relay circuit is activated by a relativechange ∆s/s of the signal s. Consequently, the detectionsensitivity of the activity response function α2 ∼ 1/s. Ina population of communicating cells, the signal strengths is proportional to the cell density N . Hence, FCD ren-ders the signal relay efficiency N independent of the celldensity N . To explain the accelerated pulsing at increas-ing cell densities, other aspects of the system need to beconsidered, e.g., cAMP clearance by phosphodiesterasesecreted by cells61. Building these details into the FHNmodel, Sgro et al.3 showed that the coupled equations areable to qualitatively reproduce the observed behaviour.The data analysis procedure illustrated by Fig. 6 offers adirect way to link pulsation from 8 min to about 6 minwith a faster signal clearance effected by a higher concen-tration of phosphodiesterase in the surrounding medium.The long firing interval at early stage of the develop-ment could be attributed to physiological differences inthe intracellular molecular network, e.g., a much longernegative feedback time τy that awaits experimental veri-fication59,60. With this type of data, a similar procedurecould be applied to analyse the segmentation clock in thepresomitic mesoderm7.

Our third example, the glycolytic oscillation in yeastcell suspensions, is also an open problem. Simulationstudies of a detailed model of yeast glycolysis55 yielded arelatively simple phase diagram shown in Fig. 5B, withthe intracellular glucose and acetaldehyde concentrationsas control parameters. As reported previously55, cells inthe white region oscillate spontaneously in a constant en-vironment, driven by an instability associated with thenegative feedback in ATP autocatalysis. In the neigh-bourhood of this region, we found that the ATP concen-tration adapts to the intracellular environment, in par-ticular to a sudden shift in acetaldehyde concentrationthat affects directly the intracellular NAD/NADH ratio.The adaptation error increases as one moves away fromthe oscillatory region. These dynamical features are cap-tured by a reduced model of ATP autocatalysis we pro-posed to approximate the low-dimensional attractor ofthe full model at high glucose concentrations. We thenconsidered a homogeneous population of cells that carryout fermentation according to the reduced model, usingmembrane permeability D to tune intracellular acetalde-hyde concentration at a given cell density. When D ismuch greater than the ACE turnover/degradation rates,the intracellular and extracellular ACE levels are equili-brated. In such a situation, collective oscillations on thelow cell density side first emerge through synchronisa-tion of individual cells that enter the self-oscillatory state.Further increase of the cell density elevates both intra-cellular and extracellular ACE levels, eventually bringsindividual cells out of the self-oscillatory state. How-ever, the population continues to oscillate following the

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DQS scenario. Slower cross-membrane diffusion drivesup intracellular ACE level and, at some point, eliminatesself-oscillation. Nevertheless, the population may stilloscillate via DQS at an intermediate range of cell den-sities. Beyond an upper critical cell density, the dimin-ishing adaptive response of glycolytic flux to the signaleventually arrests collective oscillations. This new phe-nomenon, which we name inverse DQS, is quantitativelypredicted by our theory. We note that the strong cell-to-cell variability observed in single-cell experiments56 couldsignificant alter the behaviour shown in Fig. 5G on thelow cell density side, an issue we leave to future work.

These model studies helped to refine and resolve var-ious quantitative issues in the induction of collective os-cillations in well-studied systems, and at the same timeinspire novel applications built around adaptation-drivensignal relay. One promising direction to follow is thedevelopment of artificial oscillatory systems with tech-niques from synthetic biology62–65. In analogy with thehair cell/glass fibre setup, one may think of tricking aquorum-sensing cell to oscillate by confining it to a vol-ume small enough to enable positive signal relay.

In statistical physics, the response function formalismis widely used to analyse system level response to en-vironmental perturbations, but its application to collec-tive behaviour in biological systems is still limited. Ourexamples show that cell models with different levels ofbiological detail, out of either necessity or convenience,could yield qualitatively or even quantitatively similarresponse curves with respect to, say the production ofa particular chemical used in cell-to-cell communication,which is reassuring. As these curves are increasingly ac-cessible from experiments, their direct use for analysisand hypothesis building is highly desirable. With respectto the link between adaptation and collective oscillations,our formulation unifies and generalises previous studiesin at least three specific settings. The first is an abstract3-variable model that connects fold-change detection ofindividual cells to the robustness of collective oscillationsover a broad range of cell densities4. In the second case,adaptation was proposed to play an important role inthe collective oscillation of neuronal networks49. Lastly,an Ising-type model of chemoreceptor arrays in E. coli50

predicts that increasing the coupling strength betweenadaptive receptors drives the system to collective oscilla-tions, although in reality the chemoreceptor array man-ages to operate below the oscillatory regime. Despite therisk of running into an oscillatory instability, the couplingenhances sensitivity of the array to ligand binding. Alongthis sensitivity-stability tradeoff, one may speculate thatsome of the reported collective oscillations under labora-tory conditions could actually arise from over perfectionof adaptive/homeostatic response in the natural environ-ment, a hypothesis that invites further experimental test-ing.

MethodsExtended materials and methods are presented in

Supplementary Information.

An adaptive model with cubic nonlinearity.The data presented in Fig. 3 were obtained fromnumerical integration of the coupled equations41,44:τaa = −a− c3a3 + y+α2s+ ηa, and τy y = −a− εy+ ηy.Here y is a memory node that implements negative feed-back control on a, ε sets the adaptation error, and τa andτy are the intrinsic timescales for the dynamics of a andy, respectively. ηa and ηy are gaussian white noise withzero mean and correlations: 〈ηa(t)ηa(τ)〉 = 2Tτaδ(t− τ)and 〈ηy(t)ηy(τ)〉 = 2Tτyδ(t − τ), where δ(t) is theDirac delta function. The cubic nonlinearity (c3a

3) isneeded to limit cellular activity to a finite strength. Forsimplicity, we choose α2 = 1 so that the response func-tion defined by Ra(ω) = 〈a(ω)〉/s(ω) can be comparedwith its equilibrium counterpart that satisfies the FDTR′′a = ωCa(ω)/(2T ), with R′′a denoting the imaginary

component of Ra. Data in Fig. 3 were obtained bycoupling cells via Equation (1) with F (s) = −Ks andξ = 0.

Existence of oscillatory state under an adaptiveresponse. We have shown in the Main Text that adap-tive intracellular observables exhibit a phase-leadingresponse in a certain frequency interval. For a givenadaptive observable a, the phase lead φa(ω) spans acontinuous range from 0 to a maximum value φmax

a

(< π). Meanwhile, the phase delay φs = − tan−1(ωτs)varies continuously from 0 to −π/2 [Equation (6)].Since τs controls how fast φs(ω) decreases from 0 to−π/2 as ω increases, intersection of −φs(ω) with φa(ω)can always be found by tuning τs. In particular, whenτs → 0, a solution is found at the high frequency endof the active frequency interval where φa(ω) = 0. Fromthis discussion, we see that the onset frequency ωo ofoscillations is mostly determined by the intracellulardynamics, i.e., φa(ω), but the medium can have a weakeffect on ωo when its relaxation time is comparable tothat of the intracellular dynamics.

Coupled FitzHugh-Nagumo model. A single FHNcircuit takes the form, τaaj = aj − a3j/3− yj +α2s+ ηaj

,τy yj = aj − εyj + a0 + ηyj

. The positive sign of the firstterm in the equation for aj gives rise to excitability.In the absence of the stimulus s, each cell assumes theresting state with a mean activity ars ≡ 〈aj(t)〉. Forsmall values of ε, the resting state activity ars ' −a0is nearly constant under a slow-varying s(t). FHNcircuits are coupled together through a signal field whose

dynamics is described by, τss = −s + α1

∑Nj (aj − ars).

The parameters used in generating Fig. 4 are: α2 = 1,N = 1000, ε = 0.1, T = 0.1, τa = 1, τy = 5, a0 = 1.5,and τs = 1. α1 = N/(Nα2) is determined by the controlparameter N .

Data availabilityThe data that support the findings of this study are

not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. The copyright holder for this preprint (which wasthis version posted November 24, 2019. . https://doi.org/10.1101/421586doi: bioRxiv preprint

10

available from S.-W. Wang on request. They can also begenerated from the provided code.

Code availability

The code that support the findings of this study are avail-able at https://github.com/ascendancy09/Collective-oscillations.

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AcknowledgementsThe authors thank Allon Klein and Kyogo Kawaguchi forhelpful discussions and suggestions on the manuscript.The work is supported in part by the NSFC under GrantNos. U1430237, 11635002 and U1530401, and by the Re-search Grants Council of the Hong Kong Special Admin-istrative Region (HKSAR) under Grant No. 12301514and C2014-15G.

Author contributionsS.-W. W. and L.-H. T. designed research, performed re-search, analysed data, and wrote the paper.

Competing interestsThe authors declare no competing interests.

Additional InformationSee Supplementary Information.

not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. The copyright holder for this preprint (which wasthis version posted November 24, 2019. . https://doi.org/10.1101/421586doi: bioRxiv preprint